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  "body": "\n## Definition\n\nLet $S$ be a [[scheme|5A35TYZD02]], and let $\\mathcal{C}$ be the category of schemes over $S$ with the fppf topology.\nFor an $S$-scheme $U$, let $h_U$ be its [[functor of points|SWWW9X4A3N]].\nAn *algebraic space* over $S$ is a functor $F \\colon \\mathcal{C}^{\\mathrm{op}} \\to \\mathrm{Sets}$ such that:\n\n1. $F$ is a [[sheaf|TB6TM6R4MD]] for the fppf topology.\n2. For all $S$-schemes $T_1, T_2$ and maps $h_{T_1} \\to F$, $h_{T_2} \\to F$, the [[fiber product|1DZMT4S9FB]] $h_{T_1} \\times_F h_{T_2}$ is isomorphic to $h_T$ for some scheme $T$.\n3. There are an $S$-scheme $U$ and a map $h_U \\to F$ with the following property.\nFor every $S$-scheme $T$ and map $h_T \\to F$, the induced morphism $U \\times_F T \\to T$ is surjective and [[étale|H84VHC8PX2]].\n\n## Remarks\n\nEvery $S$-scheme $X$ gives an algebraic space $h_X$.\nClause 1 holds because representable functors are fppf sheaves, clause 2 because $h_{T_1} \\times_{h_X} h_{T_2} = h_{T_1 \\times_X T_2}$, and clause 3 holds with $U = X$.\nA functor that is not a sheaf is not an algebraic space.\nFor example, the constant presheaf with value $\\set{0, 1}$ fails clause 1: on the disjoint union of two copies of $S$ it should have four sections, but it has two.\nClause 2 says that the diagonal $F \\to F \\times F$ is representable by schemes, and clause 3 then makes sense.\nSome authors use the étale topology in clause 1 and add a quasi-compactness condition on the diagonal, as in Knutson's book.\n",
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